A-level Physics Coursework - Measuring Viscosity

A2 Physics Coursework

Matt R****

Can I determine the viscosity of a fluid through experiment? The aim of my investigation was to explore the relationships between a number of variables which are described by Stokes’ Law and ultimately to use these in order to calculate the viscosity of a fluid from experimental results; my experimental values could then be compared against known ‘textbook’ values to assess the success of my measurements. With the exception of superfluids; all fluids - both liquids and gasses – exhibit the property of viscosity. The viscosity of a fluid, usually measured in Pascal-seconds (Pa·s), describes its resistance to deformation and hence the ease with which it flows. It is caused by internal fluid friction due to intermolecular forces, thus it is affected by both temperature and pressure in gasses, and just temperature in liquids. Newtonian fluids have viscosities that remain constant under constant conditions, regardless of the forces acting upon them. “Thick” liquids that flow slowly (such as glycerol) have higher viscosities, whereas “thin” liquids (like water) that flow easily have lower viscosities. There are many ways to measure viscosity; however I chose to do so using a falling-sphere viscometer as this was the most viable method to be performed in the lab using standard apparatus. Stokes’ Law is an expression (figure 1) derived by George Gabriel Stokes that describes the drag force ( ) exerted on a sphere in a Newtonian fluid. The drag force is a measure of the frictional force (or ‘resistance’) in Newtons experienced by the sphere as a result of the fluid being made to flow around it. Fig 1.

Where: is the dynamic viscosity of the fluid in Pa·s is the radius of the ball in m is the velocity of the ball in m s-1 In the form shown in figure 1, this expression was of limited use to me; I planned to take measurements by dropping ball bearings through a vertical tube containing the specimen fluid, however needs to be known to calculate the viscosity using this equation and this cannot be directly measured in this method. By looking at the forces acting on a ball when falling through the fluid at terminal velocity, another more relevant equation could be derived. When a ball is falling due to gravity there are three forces acting upon it; the force of gravity ( ) acts downwards, whilst the drag force of the fluid ( ) and the buoyancy of the ball in the fluid ( ) act in the opposite direction. At terminal velocity, the net force acting on the ball must be zero as there is no acceleration (from the equation ). Therefore the following basic statement could be made about the forces acting upon the ball when moving at terminal velocity: Fig 2.

Page 1 of 25

A2 Physics Coursework

Matt R****

The weight of the ball (i.e. gravitational force,

) can be expressed in terms of volume and density:

Fig 3.

Where: is mass in kg is the gravitational acceleration, taken to be 9.81 m s-2 and are density in kg m-3 is the radius of the ball in m The buoyancy ( ) can be considered to be the weight of the displaced fluid, therefore the same equation can be used as figure 3 but with the density of the fluid ( ) substituted in place of the ball’s density ( ). The volume of the fluid displaced will equal the volume of the sphere, and the weight of the displaced fluid will be the product of the volume and density. Figure 4 shows how all these expressions can be combined. Fig 4. The expressions for the different forces are combined as shown in figure 2. This relationship describes a ball travelling at terminal velocity (when the net force is zero), so the term for velocity in the original Stokes’ law equation now has to be made specific to terminal velocity .

is made the subject and the similar expressions for

(

and

are combined:

)

The whole expression is then divided by ( ) ( )

:

Finally it is divided by 6. Though this could be done as part of the previous step, I separated this out to make it easier to identify where the fraction of in the final equation comes from. (

(

)

)

Page 2 of 25

A2 Physics Coursework

Matt R****

This was now a much more useful form of Stokes’ law, as velocity is very easy to measure. If all the other terms in the equation were known and kept constant, the terminal velocity of the ball (determined by measuring the time taken to pass between two points) falling through the fluid could be used to calculate the fluid’s viscosity. The tidied and rearrangement form of the equation to make viscosity ( ) the subject is shown in figure 5. Fig 5. (

)

Where: is the terminal velocity of the ball in m s-1 is the radius of the ball in m is the gravitational acceleration, taken to be 9.81 m s-2 is the density of the ball in kg m-3 is the density of the fluid in kg m-3 is the dynamic viscosity of the fluid in Pa·s

Now that I understood the underlying theory behind calculating viscosity I could return to my aim and planning how I would perform my measurements. I was given access to three different fluids: water and two different types of bubble bath. I assigned each fluid a number:   

Fluid 1 was water Fluid 2 was the blue Tesco branded bubble bath Fluid 3 was the orange “Tesco Value” bubble bath

Unfortunately - but unsurprisingly - there were no known values of viscosity for the two bubble baths, so I decided that I would first perform the experiment with water and compare my calculated value for viscosity with known values (as originally planned) to verify the theory and method, then I would repeat the experiment to calculate the viscosity of the other fluids and estimate the confidence I could have in these results. The three 1.8m tall tubes were filled with the fluid a few days in advance of my measurement-taking; this was to ensure that any air bubbles trapped whilst filling them would have time to float to the surface and that enough time was allowed for the fluids to reach room temperature. The room in which the tubes were located was air-conditioned and usually maintained at an approximately constant temperature of 20.5°c. I placed the three tubes against a white wall so they would have a plain backdrop – this was to make the location of the ball easier to distinguish whilst it was falling. I later improved this further by carefully backlighting the wall behind the tubes with several lamps to provide even greater contrast with the dark balls. Because I had decided to use a camera to film the balls’ descents, it was also essential to ensure that the light from the lamps did not reflect on the front of the tubes as these bright reflective glares could mask the paths of the balls on the video.

Page 3 of 25

A2 Physics Coursework

Matt R****

Next I chose four ball bearings with differing radii; I had to ensure they were all made from the same metal so they would have the same density (see the density calculations later in this investigation). It was important that they were magnetic so that I could recover them from the bottom of the tubes after each drop by using a strong magnet. To make referencing each ball easier I gave them descriptive names based on their relative sizes – the names and radii of the balls I used can be seen in figure 6 below. I mention how I measured the balls later in this investigation. Fig 6. “Large”

–

12.50mm (1.25x10-2m)

“Medium”

–

9.50mm (9.50x10-3m)

“Small”

–

7.55mm (7.55x10-3m)

“Tiny”

–

4.77mm (4.77x10-3m)

Next I needed to devise an accurate method of measuring the velocity of the falling balls. I knew that I would need to measure the time taken for the ball to travel between two points of known separation (as suggested by the equation in figure 7), but I instantly knew that I would have issues achieving reasonably accurate results with this if measuring the time myself by operating a stopwatch / timer. Fig 7. ⁄

I had already marked a 0.5m vertical section at the bottom of each tube (two clearly marked lines drawn 50cm apart) with which to base the velocity measurements on. I deemed 0.5m to be an appropriate distance for a number of reasons: Though I later performed some preliminary tests to explore and confirm this, I had to be confident that the ball would reach terminal velocity before its velocity was measured (because the expression for calculating viscosity assumes terminal velocity, as shown in figures 2 and 4). Taking into account a 10cm space at the bottom of the tube to allow for the balls to collect at the bottom between runs without interfering with the measurement, 50cm was the furthest up the tube from this that I could mark and still remain wholly confident that the ball would be travelling terminal velocity when it entered this section. From a performing a few quick online tests and games, I knew that my average reaction time was roughly 0.2 seconds, and after briefly experimenting with my apparatus I estimated that the large ball took around half a second or less to pass through this 0.5m section at the bottom of the tube. Alone this seemed bad enough, but two reactions were to be required on my behalf for each measurement – one to start a stopwatch when the ball crossed the first mark and one to stop it as it crossed the second. Add to this that it was also difficult to judge when the fast-moving ball passed each mark, and this accumulation of error condemned this method as being ineffective for the situation. I did not have access to any light gates that would fit around the tube, and even if I had, it wasn’t very likely that the balls (the ‘small’ and ‘tiny’ balls in particular) would consistently break the narrow beams of light on each run and trigger the gates. Without access to more complex and expensive light gate equipment, this was not a viable option. Page 4 of 25

A2 Physics Coursework

Matt R****

Eventually I decided that I would use a camera to film each ball drop and calculate the time between two points by analysing the video on a frame by frame basis. Even a 25fps (frames per second) video camera would improve on the limitations of my reaction times, allowing the time to be approximated to the nearest 0.04 seconds (assuming the location of the ball was clear in each frame). A higher frame rate camera could be used to further improve on this. I had access to a 120fps camera; however the resolution was much lower, thus reducing the accuracy with which the location of a ball could be distinguished. I decided this was not a very beneficial compromise and so settled on using my hard disk camcorder which filmed with a widescreen resolution of 720x576 pixels at 25fps (interlaced). After considering the details, I realised I could exploit the interlacing of the videos capture by the camera to my advantage and effectively obtain 50fps video, increasing the accuracy of my measurement to the nearest 0.02 seconds. In an interlaced video, each frame is split into two ‘fields’ (usually named either “upper and lower”, or “odd and even” fields) which only contain half the vertical resolution of the full frame. These fields carry alternate horizontal lines of the frame, hence the naming ‘odd’ and ‘even’ (if the lines were to be numbered). See figure 8 below for a basic diagram of a full frame being split into alternate fields. Fig 8. Full frame

Upper field

Lower field

Figure 8 is a very simplified representation of interlacing in which there is no motion and so it can be said that both fields are combined to form a full frame. However, when motion is involved, the content of each field is captured at slightly different times and is independent from its other field; each field can no longer be thought of as half of a full frame, but is effectively its own frame (just with half the vertical resolution). This is essentially the purpose of interlacing – it has origins in CRT (cathode ray tube) line-based displays and was intended as a way to increase the ‘refresh rate’ (analogous though not always directly interchangeable with ‘frame rate’) of a video signal without increasing the bandwidth required to carry it. Though each field does not contain the full vertical resolution, this is not perceived by the human eye and instead is seen as smooth video when played back at full speed. By filming the ball drop with the camera rotated through 90° so that the path of the falling ball would be parallel to the interlacing lines, the reduced vertical resolution would not impede my ability to pinpoint the location of the ball on each frame; this was assuming that the smallest ball would be wide enough to span a width of at least several pixels when captured on the video, which it easily would do and so this didn’t present a problem. Detail in horizontal motion of the ball (represented by the vertical resolution of the camera) was not so important to my investigation, except possibly to indentify and examine drops where the ball interacted with the edges of the tube and was subsequently slowed down.

Page 5 of 25

A2 Physics Coursework

Matt R****

Conversely, the full horizontal resolution of each field would be orientated such that detail in the vertical motion of the ball remained the same as it would be if the video were not deinterlaced. Fig 9.

So if I broke the interlaced video of my camcorder down into individual fields, I could in effect double the frame rate to 50fps. A horizontal resolution of 720 pixels gave the potential of tracking the vertical motion of the ball within a matter of millimetres, depending mainly on the framing of the camera shot. Figure 9 shows sequential stills taken every 2 frames from a test video of the large ball dropping through the marked 50cm section at the bottom of the tube containing fluid 2, demonstrating the level of accuracy that could be achieved in following the motion of a ball using this method. Fig 10.

At this point, I judged that my method posed no significant hazards or risks. There was a higher risk of the tube containing water being knocked over as this had a less sturdy/stable base; however I ensured that the tubes were located up against a wall where they were unlikely to be accidentally knocked. Unintentional ingestion of the fluids (for example, if I had got them on my hands then eaten food without washing my hands first) would have been unpleasant but not dangerous. The balls could have presented a trip hazard if they dropped onto the floor, but I kept them in a plastic container when not in use. I also had to ensure that the magnet I used to remove the balls from the tube did not get placed close to my camcorder, which could have corrupted data on the hard disk drive that the videos are stored on. Page 6 of 25

A2 Physics Coursework

Matt R****

Before making any further adjustments to the method, I first needed to verify that the balls reached terminal velocity for the section of tube that I had marked out. These preliminary runs of the measurement method also allowed me to test out the processing of the videos on the computer. To decompress, deinterlace and trim the videos I used free video editing software named VirtualDub. I used filters to break the video into its individual fields, and then resized these to match the original aspect ratio. I then imported the AVI files saved from VirtualDub into an open source Java program called Tracker (see figure 9). Tracker allows you to set of scaled axes over a video and then track the path of an object moving relative to these axes simply by plotting the location of the object on each frame. The data could then be exported in a comma-separated format that can then be opened in Microsoft Excel for further analysis. Fig 11.

Test drop 1 (fluid 2, large ball)

Displacement (m)

0.6

0.4

0.2

0 0

0.1

0.2

0.3

0.4

Time (sec)

Figure 11 shows the results from a test drop. On a graph of displacement plotted against time, the gradient equals the velocity, so the linearity of the graph represents a constant velocity and thus it can be assumed that the ball is travelling at terminal velocity.

Following this, the first modification I made was to check the tubes were vertical using a spirit level and to adjust bases of the tubes to make corrections where necessary. Whilst it was impossible to prevent altogether without having a much wider tube, this was one step I could take to try and reduce the probability of the balls interacting with the edges of the tubes as they fell (which could affect their velocity).

When setting up the camera I realised that its position relative to the tube could have a significant effect on the results due to perspective and parallax errors. Page 7 of 25

A2 Physics Coursework

Matt R****

The first of this type of error I looked at is illustrated in figure 12 – whilst the two pairs of points along the line are equally spaced, from the marked point-of-view; points B will appear much closer together than points A. In the context of the experiment, if the tube were to be filmed from a high angle the ball may erroneously appear to decelerate (as the perceived distance it travels over a set period of time appears to decrease). Conversely, if filmed from a low angle, the ball may appear to accelerate when it is in fact moving at constant velocity. This kind of error could seriously affect the accuracy of my measurements. Fig 12. Two pairs of equally spaced points are marked on a ruler. When viewed from an oblique angle they appear farther apart.

viewpoint

Viewed at an angle Viewed almost head-on

I decided to perform a few more test drops to confirm this idea. The graphs in figures 13 and 15 show the data obtained from four test drops; all were filmed the same distance from the tube, but for two the camera was positioned at a very high angle relative to the tube, and the other two filmed from the floor. Figure 13 which displays the data from the high angle drops clearly indicates a slight curve in the trend which relates to a measured deceleration (the gradient of the graph equals the velocity, so a lessening in the steepness of the gradient indicates a slower velocity). This deceleration did not in fact happen – if anything the ball should accelerate - but is a result of the error introduced by filming from a high angle. Fig 13.

High Angle Test Drops 200 180 160

Distance (cm)

140 120 100

Large Ball

80

Medium Ball

60 40 20 0 0

0.2

0.4

0.6

Time (sec)

Page 8 of 25

0.8

1

A2 Physics Coursework

Matt R****

To present this in a different way, I plotted a graph of velocity against time for the drop of the large ball. The gradient of this graph represents the acceleration – a positive gradient would signify acceleration, no gradient would mean that the velocity was constant, and a negative gradient would indicate deceleration. As can be seen in the graph below (figure 14), there was a clear negative gradient which showed that the ball appeared to decelerate over the course of the drop. Fig 14.

High angle test drop, v/t graph 3.0

Velocity (ms-1)

2.5 2.0 1.5 1.0 0.5 0.0 0.0

0.2

0.4

0.6

0.8

1.0

Time (sec)

As was expected, the opposite effect was observable in the drops filmed from a low angle (figure 15). Fig 15.

Low Angle Test Drops 140 120

Distance (cm)

100

80 Large Ball 60

Medium Ball

40 20 0

0

0.2

0.4 Time (sec)

Page 9 of 25

0.6

0.8

A2 Physics Coursework

Matt R****

Parallax error could also affect the moment when the ball is judged to be crossing a mark on the tube. Figure 16 shows how a ball falling past a mark appears to be lined up with the mark at very different points depending on whether the tube is viewed from A, B or C. Refraction of light at the medium boundaries (fluid, tube, air) could also introduce similar errors if the marks on the tube were viewed at an angle other than head-on. Fig 16. At viewpoint A, the ball appears to be lined up with mark after it actually crosses the mark

A single ball is dropped through a marked tube and observed from three different angles.

When viewed head on from viewpoint B, the crossing of the mark can be correctly identified

At viewpoint C, the ball appears to be in line with the mark before it crosses it

It crossed my mind that it was possible to accurately compensate for the first type of error using trigonometry if the relevant distances and angles between the camera and tube were known, however this would have introduced serious complication intro the processing and analysis of the videos. The Tracker software is only intended for analysing videos with a constant scale applying across the whole frame. I decided it was best to try and reduce these errors as much as possible through the positioning of the camera before taking any measurements. Each of these errors depended on the angle from which the tube or mark was viewed and ideally would be viewed head-on. However the ball had to be observed crossing and travelling between two marks, and it would be a physical impossibility to position and frame the camera shot such that both marks were filmed head-on! The only solution was to increase the distance between the tube and the camera as much as possible, thus reducing the angle that each mark was viewed from. Figure 17 illustrates this. Fig 17. Camera close to tube The camera is elevated and positioned such that it points horizontally to an imaginary central point between the two marks.

Camera further from tube A greater distance between the camera and the tube reduces the angle at which both marks on the tube are viewed from.

50 cm marked section

camera

Page 10 of 25

A2 Physics Coursework

Matt R****

These two stills (fig 18) were taken from test videos and show the significant difference caused by the distance between the camera and a marked section of tube – the first was filmed 3 metres away whilst in the second the camera was placed just 85 cm away from the tube. Fig 18.

Marks are viewed from a very shallow angle.

Curvature of tube is more apparent

Greater angle between the camera’s horizontally central ‘horizon’ and the marks.

3.00m

0.85m

The horizontal centreline of both images was marked as a reference point. The bottom tape marking is to be ignored as this piece of tape was not applied to the tube straight to being with. In the first image, the edges of the top tape marking appeared very straight and horizontal. In the second, the view of the tape marking exaggerated the curvature of the tube due to the greater angle between the camera lenses’ midpoint and the tape marking. The perspective seen in the first image was much more desirable for taking accurate measurements in my investigation. Eventually I settled on filming the ball drops with a distance of 3m between the tube and the camera. The camera would be elevated off the ground such that it would lie on an imaginary line drawn perpendicular from the midpoint of the 50cm marked section of the tube (as shown in figure 17) – taking into account the base/stand of the tube and the small segment below the marked section, this usually meant having the camera at an elevation of roughly 35cm. I also checked that the camera was horizontal using a spirit level. As discussed previously, the camera was rotated through 90 when filming the drops such that the horizontal reference of the camera was parallel to the vertical tube. Once the camera was set up to these ‘specifications’, it was usually already lined up with the marked section and the shot could be framed simply by adjusting the zoom such that the marked area filled most of the frame. I could tell when the camera was positioned properly because the top and bottom markings lay

Page 11 of 25

A2 Physics Coursework

Matt R****

equal distances from the top and bottom of the frame respectively without any adjustment to the angle of the camera needed. Occasionally however, some adjustment was needed – this was due to the camera slipping in the clamp of the stand I had made to hold it. The camera was held to a sturdy stand using a single clamp with some protective polystyrene padding between the clamp and the camera in order to protect the camera. The thick polystyrene allowed me to close the clamp more tightly around the camera as the pressure from the small metal clamp ‘fingers’ (which would otherwise have damaged the camera) was spread across a larger surface area and, within reason, the polystyrene would protect the camera from small over tightening of the clamp by deforming and compressing under the extra pressure before the camera would. To avoid knocking the camera out of position when starting and stopping the recording (and consequently having to carefully reposition it), I used the infra-red remote control that came with the camera. Lastly I considered the effect of air bubbles on my results – dropping the balls such that they were not released whilst touching the surface of the fluid would result in a significant amount of air being trapped on the underside of the ball for the duration of its descent. This would almost certainly have an effect on the velocity of the ball for a number of reasons, including: the fluid resistance depends on the form (shape) of the object passing through the fluid and the air would alter this; the air has a different density to the ball; the air would cause turbulence in the fluid. Also, Stokes’ Law only applies to spherical objects. Once a ball with air trapped underneath it reached the bottom of the tube, the air would rise back up the tube as a bubble, spreading smaller air bubbles throughout the fluid as it descended; these would possibly have an effect on any subsequent ball drops as they would affect the continuity of the fluid. I filmed a large number of test drops to investigate the effect of air bubbles in the fluids on the falling balls, but the interactions and behaviours I observed appeared to be very inconsistent and unpredictable. I guessed that the theory and modelling of these interactions would be extremely complex and not worth trying to pursue any further, especially considering that there would be no benefit for this investigation in greater understanding of them, only in knowing that they occurred and that it was best to try and minimize their incidence. I concluded that I would try to release each ball from such a position that it was submerged in the fluid at the top of the tube, and only do so once I had checked that there was no air bubbles trapped on the surface of the ball.

As well as my velocity measurements, I also needed to know the values for some of the other terms in the equation I derived (figure 5). These included the diameter of each different ball, the density of the balls, and the density of the three fluids used. These can be seen in the tables below (figures 5 and 6). I made measurements of the diameter (D) of each ball using digital veneer callipers that gave readings to the nearest 0.001m. Spherical objects can be difficult to accurately measure with callipers, so to ensure that I was always measuring the full diameter of each ball I would first roughly close the callipers on the ball then check this was the largest part of the sphere by trying to pass it between the jaws – if it couldn’t pass between the separated jaws then I knew that I hadn’t quite measured full the diameter. Usually, if this were the case, then carefully pushing the ball through would widen the jaws to the correct width.

Page 12 of 25

A2 Physics Coursework

Matt R****

Using the diameter measurements I then calculated the volume (V) of each ball using the equation for the volume of a sphere shown as shown in figure 19 (where the radius in metres, R, is half of D). Fig 19.

The mass measurements (M) were made using a set of digital balances with a precision of 0.05g. Three measurements were made for each ball and then a mean value (m ̅ ) calculated. From all these measurements (shown in figure 20), the average density of the four balls could be calculated. The equation for density can be seen in figure 21. Fig 20. Ball

D(m)

V (m³)

m 1 (kg)

m 2 (kg)

m 3 (kg)

m ̅ (kg)

Density (kg m¯³)

Large

2.50x10-2 1.90x10-2 1.51x10-2 9.53x10-3

8.18x10-6 3.59x10-6 1.79x10-6 4.53x10-7

6.38x10-2 2.80x10-2 1.40x10-2 3.50x10-3

6.39x10-2 2.82x10-2 1.40x10-2 3.50x10-3

6.38x10-2 2.81x10-2 1.40x10-2 3.60x10-3

6.38x10-2 2.81x10-2 1.40x10-2 3.53x10-3

7.81x103

Medium Small Tiny

Average Density:

7.82x103 7.81x103 7.80x103

7.81x103

Fig 21.

From the lack of variation in the densities I could draw the assumption that all the balls were made from the same metal. The very slight differences were likely to be caused by defects in the balls (such as scratches in the surface) or small errors in the measurements. These were possibly emphasised when rounding these measurements (and the subsequent calculated density results) to three significant figures. For the purposes of the investigation I treated all the balls as sharing the same density of 7.81x103 kg m-3. I also needed to know the densities of the fluids I was to be testing (values for known volume of each fluid.

). I did this by weighing a

Fig 22. Fluid tested Fluid 1 Fluid 2 Fluid 3

Mass including container (kg) 0.1175 0.1196 0.1198

Mass of fluid (kg) 0.0998 0.1019 0.1021

These measurements were made under the following conditions: Mass of measuring container (kg): Volume of fluid (m³): Temperature (°c):

0.0177 0.0001 20.6 Page 13 of 25

Density (kg m¯³) 998 1019 1021

A2 Physics Coursework

Matt R****

Now that I had developed and improved my method, I was ready to carry out my final set of measurements. I chose to make a total of 16 velocity measurements for each fluid; 4 drops for each of the 4 balls. This large number of repetitions for each fluid would hopefully increase the accuracy with which I could determine their respective viscosities. It was a lot easier and more efficient to process the videos in large batches, so I chose to perform all my measurements in one session and analyse them afterwards. This would also allow me to ensure that affecting factors such as temperature remained close to constant throughout all the measurements. I encountered very few difficulties when taking my measurements – I chose to discard some drops that were visibly turbulent or appeared to be affected by air bubbles, but mostly the balls appeared to drop smoothly and consistently. Completing my measurements took slightly longer than expected due to the time I had to spend setting up my camera for each fluid, and in the end I utilised the entire remainder of my allocated lab time. Once I had analysed all my videos, I imported the distance/time data for each ball drop from Tracker into Microsoft Excel, where I plotted a graph of this data for each fluid and ball size. I allowed Excel to plot its own linear trend lines on the charts, and the gradient of each of these lines was equal to the terminal velocity of the ball for that drop. The different runs had varying y-intercepts which were simply due to the way in which the axes were setup up and the ball tracked in Tracker, and would not have any effect on the results as it was only the gradient that was of importance. The first set of results I analysed were those for fluid 1, water...

Fluid 1, tiny ball Run 1

Run 2

Run 3

Run 4

0.6

v1 = 1.00 m/s v2 = 1.24 m/s v3 = 1.28 m/s v4 = 1.01 m/s

Distance (m)

0.4

0.2

0.0 0.0

0.1

0.2

0.3 Time (sec)

Page 14 of 25

0.4

0.5

A2 Physics Coursework

Matt R****

Fluid 1, small ball Run 1

Run 2

Run 3

Run 4

0.6

v1 = 1.43 m/s v2 = 1.50 m/s v3 = 1.31 m/s v4 = 1.29 m/s

Distance (m)

0.4

0.2

0.0 0.0

0.1

0.2

0.3

0.4

0.3

0.4

Time (sec)

Fluid 1, medium ball Run 1

Run 2

Run 3

Run 4

0.6

v1 = 1.56 m/s v2 = 1.62 m/s v3 = 1.57 m/s v4 = 1.55 m/s

Distance (m)

0.4

0.2

0.0 0.0

0.1

0.2 Time (sec)

Page 15 of 25

A2 Physics Coursework

Matt R****

Fluid 1, large ball Run 1

Run 2

Run 3

Run 4

0.6

v1 = 1.66 m/s v2 = 1.62 m/s v3 = 1.61 m/s v4 = 1.62 m/s

Distance (m)

0.4

0.2

0.0 0.00

0.10

0.20

0.30

0.40

Time (sec)

At a glance I was relatively happy with how these results looked; the velocity measurements of the four runs for each ball size all seemed to agree with each other (had similar gradients), more so was the case in the larger balls – I observed that the range of the velocity measurements for each ball decreased as the size of the ball increased, from a range of 0.28ms-1 for the tiny ball to just 0.04ms-1 for the large ball. From these results I plotted a graph of Vt against R2. I chose to do this as it was a relationship implied by the equation in figure 5; if the experimental data conformed to the theory then Vt should be proportional to R2 and result in a linear trend passing through the origin when plotted. The gradient (G) of this graph could then be used to calculate a value for the viscosity of the fluid using the equation at the bottom of figure 23. Fig 23. (

)

(

)

This was a method of combining all the data from the 16 ball drops in order to calculate the viscosity. Unfortunately, the plotted data for water did not fit the relationship that I expected. See graph “Vt against R2, fluid 1“ at the end of the investigation. Page 16 of 25

A2 Physics Coursework

Matt R****

There was some linearity between the values of the tiny, small and medium sized balls; however this did not pass near the origin, nor could it be extended to include the large ball. I attempted to estimate an approximate value for the viscosity based on the gradient of a line drawn through the first three data sets, however the value I calculated of around 4.0 Pa·s, which differed from the textbook value for water at room temperature (roughly 1.0x10-3 Pa·s) by such a large amount that I concluded this data set was of little use except for determining the source of the error in my results. By substituting a known value for the viscosity of water into the equation I could calculate the theoretical linear gradient for my Vt against R2 graph – this revealed that the gradient should be several orders of magnitude greater than the kind of relationship I was witnessing on my graph; this showed that the balls were falling a lot slower than be expected, indicating that other forces were acting upon them. I later hypothesised that a possible cause for this was associated to the relatively small diameter of the tube – the equations for Stokes’ Law assume an ideal, though impossible to obtain, infinite body of fluid for the object to fall through, yet I used cylindrical tubes I used with fairly narrow diameter of 5.5cm. The compressibility of liquids is very small and generally negligible so they are usually treated as incompressible. In an ideal volume of fluid, the fluid displaced by a ball as it falls can simply move around sides of the of the ball to occupy the space above the ball, which in turn ‘pushes’ the fluid that was at the sides of the ball further outwards – this is shown in figure 24. An infinite volume of fluid is theoretically ideal for modelling the flow of an incompressible fluid as the effects of this horizontal displacement will continue indefinitely throughout the entire body of the fluid. Fig 24.

source: http://en.wikipedia.org/wiki/File:Stokes_sphere.svg However, in the context of my experiment, this flow is constricted by the dimensions of the tube; the bottom of the tube is sealed so the displaced fluid still has to move around the edges of the sphere to occupy the space above it; however there is a limited cross-sectional area around the ball through which this displaced fluid can flow. The velocity of the upwards flow around the edges of the ball has to increase. This has the overall effect of slowing the ball and reducing the terminal velocity that the ball can reach. This effect should become much greater as the radius of the ball increases (more correctly, as the ratio between the radius of and tube to the radius of the ball decreases), as a relatively larger volume of fluid would be displaced and have to flow through a smaller space. Next I moved onto plotting the data for the second fluid. Page 17 of 25

A2 Physics Coursework

Matt R****

Fluid 2, tiny ball Run 1

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v1 = 0.24 m/s v2 = 0.24 m/s v3 = 0.23 m/s v4 = 0.26 m/s

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Fluid 2, small ball Run 1

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v1 = 0.97 m/s v2 = 0.98 m/s v3 = 0.93 m/s v4 = 0.99 m/s Distance (m)

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Page 18 of 25

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A2 Physics Coursework

Matt R****

Fluid 2, medium ball Run 1

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0.6

v1 = 1.19 m/s v2 = 1.24 m/s v3 = 1.40 m/s v4 = 1.17 m/s

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Fluid 2, large ball Run 1

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v1 =1.28 m/s v2 = 1.51 m/s v3 = 1.40 m/s v4 = 1.41 m/s

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Page 19 of 25

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A2 Physics Coursework

Matt R****

Again the data did not closely fit the Vt  R2 relationship as I had hoped, however it appeared to be more useful than the data from fluid 1; I could draw a curve of best fit through all the points that looked as if it would pass close to the origin should it be extrapolated. The slope of this curve decreased as R2 increased. I postulated that the curve was due to the effect I discussed on page 17 – the larger the radii of the ball (and thus the larger R2 is), the greater the strength of this effect and so the more the terminal velocity of the ball was reduced from its hypothetical “true” value. One could visualise where the theoretical linear trend might pass, and how the actual line curved further away from this line as R2 increased. I postulated that, as they would have been much less affected by this error, I could produce an estimate of the viscosity by drawing a line, roughly a tangent to the curve, which passed through the points for the two smallest balls. I did so, as can be seen on the chart “Vt against R2, fluid 2“, and calculated an estimated viscosity of the fluid of 0.70 Pas based on the gradient of this tangent and the calculation in figure 25. Fig 25. (

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Lastly was the data for fluid 3.

Fluid 3, tiny ball Run 1

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v1 = 0.31 m/s v2 = 0.34 m/s v3 = 0.33 m/s v4 = 0.31 m/s

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Page 20 of 25

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A2 Physics Coursework

Matt R****

Fluid 3, small ball Run 1

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v1 = 0.91 m/s v2 = 0.84 m/s v3 = 0.92 m/s 0.4 Distance (m)

v4 = 0.92 m/s

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Fluid 3, medium ball Run 1

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v1 = 1.16 m/s v2 = 1.17 m/s v3 = 1.11 m/s v4 = 1.20 m/s

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Page 21 of 25

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A2 Physics Coursework

Matt R****

Fluid 3, large ball Run 1

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v1 = 1.35 m/s v2 = 1.43 m/s v3 = 1.41 m/s v4 = 1.52 m/s

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The chart “Vt against R2, fluid 3“ based on this data shared a lot of similarities with the chart for fluid 2, though the range of velocities for each ball size appeared less than that for fluid 2 – there was slightly more consistency in the measurement of the terminal velocities. The line of best fit also passed closer to the origin. Again I calculated an estimate of the viscosity, as can be seen in figure 26 below. Fig 26. (

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I had now estimated the viscosity of both fluids 2 and 3; however I could have little confidence in the accuracy of these results. I had originally planned to compare my results for fluid 1 (water) with known values in order to verify the method, but my results for fluid 1 were not even suitable to gain a reasonable ‘ball-park’ estimate of viscosity from, thus I could not have any certainty that my method was valid. I identified a number of limitations in my investigation. Firstly there was the error related to the relatively small diameter of the tube restricting the flow of fluid around the ball that I identified and briefly explored on page 17 – I believe this had a significant effect on my results (reducing the terminal velocity, with greater effect with the larger balls) and was one of my main sources of error. I later discovered and researched another possible major source of error. The falling ball can experience both viscous drag – the type described by Stokes’ Law – and inertial drag, which does not depend on viscosity. The proportions of these forces being exerted on the ball are described by its Reynolds number. Page 22 of 25

A2 Physics Coursework

Matt R****

“In fluid mechanics, the Reynolds number (Re) is a dimensionless number that gives a measure of the ratio of inertial forces to viscous forces and consequently quantifies the relative importance of these two types of forces for given flow conditions.” Source: http://en.wikipedia.org/wiki/Reynolds_number In the context of this investigation, the Reynolds number could be calculated using the equation outlined in figure 27. As both the radius of the ball and the viscosity of the fluid are terms in the equation, the Reynolds number would be different for each ball-size/fluid combination in my investigation. Fig 27.

Where: is the density of the fluid in kg m-3 is the dynamic viscosity of the fluid in Pa·s is the diameter of the ball in m is the velocity of the ball in m s-1 A condition of Stokes’ Law is that it is only applicable when the Reynolds number is less than 1, though this would ideally be less than 0.1 as this is associated with completely laminar flow; when the fluid flows smoothly without turbulence, as if in layers (as depicted by the streamlines on the diagram in figure 25). When the Reynolds number is higher, inertial drag (which does not depend on viscosity) become more dominant than viscous drag and so Stokes’ Law becomes much less relevant. According to this equation, a ball falling with a greater velocity is more likely to experience turbulence and inertial drag, as is one falling through a less viscous fluid (which would also allow it to fall faster). I could not calculate the Reynolds numbers for fluids 2 and 3 as I was not confident in the validity of my viscosity estimates; however I could calculate the hypothetical values for the four balls using the averages of my measured velocities, my measured density and the textbook value for the viscosity of water at 20°c. Fig 28.

All of these were a lot larger than the Re